Effective action for bosonic string theory with enhanced symmetry

Question

See these lecture http://members.ift.uam-csic.es/auranga/lect7.pdf page 17.

Usually one derives the effective action from the massless states calculating amplitudes, otherwise through beta functions(worldsheet conformal invariance). One obtains a effective field theory containing a metric, a Kalb-ramond field and a dilaton. These came from the $N=1,\bar{N}=1$ sector of the mass spectrum. This is wellknown.

If the 25th dimension is compactified with $R=\alpha'^{1/2}$ then another massless 25dimensional fields emerge. This happens in the $N=0,\bar{N}=1$ sector and $N=1,\bar{N}=0$ sector. After analysing the enhanced symmetry from the 25d point of view one can deduce that these are $SU(2)\times SU(2)$ gauge bosons.

On page 17 of the lecture the autor says: it is possible to cook up a new 25d effective field theory by including by hand the new massless modes.

How this action looks like? Is there any paper or reference which explains the calculations or steps to obtain the action?

Answer 1

The value $R=\alpha^{\prime 1/2}$ is the self-dual radius under T-duality. One may indeed extract the massless spectrum – the spectrum of all fields much lighter than $\alpha^{\prime -1/2}$.

Because the CFT has an $SU(2)\times SU(2)$ symmetry, as can be seen from the OPEs of the currents, the spacetime physics has this symmetry, too. Because one finds (spacetime) Lorentz vector states in the adjoint of $SU(2)\times SU(2)$, it is clear that this group is the gauge symmetry of the spacetime physics.

And indeed, one may verify that the tree-level scattering amplitudes for all the relevant string modes agree with the scattering amplitudes extracted for quanta of fields in the effective action that is (a bit schematically, especially when it comes to the parts unrelated to the enhanced gauge symmetry) $$ S =\int d^{25}x\,\exp(2\phi) [R + (\partial_{[\lambda} B_{\mu\nu]})^2 + (\partial_\mu \phi)^2 -\frac 14 {\rm Tr}(F_{\mu\nu}F^{\mu\nu}) ] $$ So it is a 25-dimensional action because we ignore the 1 compactified dimension whose radius is stringy (dimensional reduction). In this 25-dimensional spacetime, there is the dilaton, the metric, the B-field, and an $SU(2)\times SU(2)$ gauge field, and they have more or less the expected terms in the effective action.

See Polchinski's Volume 1 from page 242 to 250+ or so. The effective action is probably not written there explicitly. However, you may find the 26D effective action for the uncompactified bosonic string theory on the top of page 114, reduce the dimension, and add the $SU(2)\times SU(2)$ Yang-Mills field, more or less getting the exact answer. The "Cartan" $U(1)\times U(1)$ part of the Yang-Mills action comes from the Kaluza-Klein $U(1)$ symmetry of the circle and from the components of the B-field $B_{\mu,25}$. This is "enhanced" by the extra "accidentally massless" states to the non-Abelian group.

Between equations 3.7.15 and 3.7.20 or so, Polchinski takes a different but ultimately equivalent strategy to derive the spacetime action. He derives the equations of motion from the requirement of the conformal symmetry on the world sheet (vanishing beta-functions etc.) and verifies that the same equations follow as the Euler-Lagrange equations from the spacetime action he "guesses" and "refines".